Finite Group Databasebeta

← #24113

Subgroups — #24113

Order 256.

561 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3491C211yesmaximal, characteristic
2#2853C212yesmaximal, complemented
3#2296C211yesmaximal, characteristic, complemented
4#2062C212yesmaximal, complemented
5#2044C212yesmaximal
6#1899C212yesmaximal, complemented
7#1867C214yesmaximal, complemented
8#1820C211yesmaximal, characteristic, complemented
9#552—24no—
10#551—24no—
11#543C2211yescharacteristic
12#539—22no—
13#534C2211yescharacteristic
14#529—22no—
15C2×#123C2212yes—
16#518C2212yes—
17C2×#117C2211yescharacteristic
18C2×#117C2211yescharacteristic
19C22×C42C2211yescharacteristic
20C2×#99C2214yes—
21#389—22no—
22#388—24no—
23#387—24no—
24#386C2214yes—
25#385C2214yes—
26#385C2212yes—
27#385C2212yes—
28#381C2212yes—
29C4×C4⋊C4—22no—
30#342C2212yes—
31#342C2214yes—
32#342C2212yes—
33C22×D4C2312yes—
34C23×C4C2312yes—
35C23×C4C2311yescharacteristic
36#126—24no—
37#126—24no—
38#125—22no—
39#125—24no—
40#124—22no—
41#123—24no—
42#123—22no—
43#123—22no—
44#123—24no—
45#123—22no—
46C22⋊Q8—24no—
47C22⋊Q8—22no—
48C22⋊Q8—24no—
49#121—22no—
50#120—22no—
51C4×Q8—22no—
52C4×D4—22no—
53C4×D4—24no—
54C4×D4—24no—
55#117—24no—
56#117—24no—
57#117—22no—
58#117—24no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4D412yes—
64C2×C4⋊C4D414yes—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4D414yes—
68C2×C22⋊C4C2314yes—
69C2×C22⋊C4C2311yescharacteristic
70C2×C22⋊C4C2311yescharacteristic
71C2×C22⋊C4C2312yes—
72C2×C22⋊C4C2312yes—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42—24no—
76C2×C42—21no—
77C2×C42D412yes—
78#99—28no—
79#95—22no—
80#95—22no—
81#95—24no—
82#95—24no—
83#95—24no—
84C24C2×D412yes—
85C24C2411yescharacteristic
86C2×Q8—22no—
87C2×D4—22no—
88C2×D4—22no—
89C2×D4—24no—
90C2×D4—24no—
91C2×D4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—24no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—24no—
98C22×C4—24no—
99C22×C4—44no—
100C22×C4C2×D412yes—
101C22×C4C2×D414yes—
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D412yes—
105C22×C4C2×D412yes—
106C22×C4C2×D414yes—
107C4⋊C4—24no—
108C4⋊C4—24no—
109C4⋊C4—24no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—24no—
113C4⋊C4—24no—
114C4⋊C4—24no—
115C4⋊C4—48no—
116C4⋊C4—44no—
117C4⋊C4—44no—
118C22⋊C4—24no—
119C22⋊C4—24no—
120C22⋊C4—22no—
121C22⋊C4—22no—
122C22⋊C4—22no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—24no—
126C22⋊C4—24no—
127C22⋊C4—24no—
128C22⋊C4—24no—
129C22⋊C4—48no—
130C22⋊C4#4114yes—
131C42—24no—
132C42—24no—
133C42—22no—
134C42—22no—
135C42—42no—
136C42—42no—
137C23—22no—
138C23—24no—
139C23—42no—
140C23—44no—
141C23C22×D411yescharacteristic
142C23C22×D412yes—
143C23C2×#4112yes—
144C232−1+412yes—
145Q8—82no—
146D4—42no—
147D4—42no—
148D4—44no—
149D4—82no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—28no—
155C2×C4—28no—
156C2×C4—24no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—24no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—44no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—44no—
170C2×C4—42no—
171C2×C4—44no—
172C2×C4—44no—
173C2×C4—44no—
174C2×C4—84no—
175C2×C4#12012yes—
176C2×C4#12012yes—
177C22—22no—
178C22—24no—
179C22—42no—
180C22—42no—
181C22—42no—
182C22—44no—
183C22—84no—
184C22C2×#12011yescharacteristic
185C22#52511yescharacteristic
186C22#53611yescharacteristic
187C22#54714yes—
188C4—22no—
189C4—22no—
190C4—44no—
191C4—42no—
192C4—44no—
193C4—42no—
194C4—44no—
195C4—42no—
196C4—84no—
197C2—21no—
198C2—24no—
199C2—21no—
200C2—82no—
201C2#240611yescharacteristic
202C2#302512yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC24C24
prime radical#534C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #2853C2semidirect
3. #2296C2semidirect
4. #2062C2semidirect
6. #1899C2semidirect
7. #1867C2semidirect
8. #1820C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#34912
#28532
#22962
#20622
#20442
#18992
#18672
#18202